The blue-eyed islanders
On an island live perfectly logical people. Exactly 100 of them have blue eyes, and the others have brown eyes. Everyone sees everyone else's eyes but not their own, and nobody ever talks about eye colour. Anyone who works out that their own eyes are blue leaves the island at midnight that same night, and everyone sees who has left.
One day a visitor announces, in front of everyone: “At least one of you has blue eyes.”
On which night after the announcement do the blue-eyed islanders leave? (The night of the announcement is night 1.)
Fractions, decimals with at least 3 significant digits, and expressions such as sqrt(2)/2, e^-1 or pi/4.
Suppose there were only one blue-eyed islander. Then two.
Prove by induction: if there are blue-eyed islanders, they all leave on night .
h
s Shown after a correct answer. Revealing it first marks the problem as seen, not solved.
Idea. Induction on the number of blue-eyed islanders: with of them, they all leave on night . Each blue-eyed islander sees others, waits to see whether those leave on night , and concludes from their staying that there is one more blue-eyed person, themselves.
One blue-eyed islander. They see no blue eyes, so after the announcement they know the blue-eyed person is themselves, and leave on night 1.
Induction step. Suppose the claim holds for , and there are blue-eyed islanders. Each of them sees blue-eyed people. If their own eyes were brown, those would be the only blue-eyed islanders, and by the induction hypothesis they would all leave on night . When night passes and nobody leaves, each blue-eyed islander concludes that their own eyes are blue, and all leave on night . (Brown-eyed islanders see blue-eyed people and expect them to leave on night , so they learn nothing about themselves until then.)
Answer. With 100 blue-eyed islanders, they all leave on night 100.
The puzzle inside the puzzle. With 100 blue-eyed people, everyone already knew that at least one has blue eyes. What the visitor adds is common knowledge: everyone now knows that everyone knows that everyone knows, and so on, 100 levels deep. That is what starts the countdown.